Definition

Since Diff does not have all pullbacks, to ensure that this definition makes sense, one needs to ensure that the space Mor(X)×s,tMor(X)Mor(X) \times_{s,t} Mor(X) of composable morphisms is an object of Diff. This is achieved either by adopting the definition of internal groupoid in the sense of Ehresmann, which includes as data the smooth manifold of composable pairs, or by taking the conventional route and demanding that the source and target maps s,t:Mor(X)→Obj(X)s,t : Mor(X) \to Obj(X) are submersions. This ensures the pullback exists to define said manifold or composable pairs.

Terminology

Originally Lie groupoids were called (by Ehresmann) differentiable groupoids (and also one considered differentiable categories). Sometime in the 1980s there was a change of terminology to Lie groupoid and differentiable stacks. (reference?)

Specialisations

One definition which Ehresmann introduced in his paper Catégories topologiques et catégories différentiables (see below) is that of locally trivial groupoid. It is defined more generally for topological categories, and extends in an obvious way to topological groupoids, and Lie categories and groupoids. For a topological (resp. Lie) category XX, let X1isoX_1^{iso} denote the subspace (resp. submanifold) of invertible arrows . (This always exists, by general abstract nonsense - I should look up the reference, it’s in Bunge-Pare I think - DR)

Definition

A topological groupoid X1⇉X0X_1 \rightrightarrows X_0 is locally trivial if for every point p∈X0p\in X_0 there is a neighbourhood UU of pp and a lift of the inclusion {p}×U↪X0×X0\{p\} \times U \hookrightarrow X_0 \times X_0 through (s,t):X1iso→X0×X0(s,t):X_1^{iso}\to X_0 \times X_0.

Clearly for a Lie groupoid X1iso=X1X_1^{iso} = X_1. It is simple to show from the definition that for a transitive Lie groupoid, (s,t)(s,t) has local sections. Ehresmann goes on to show a link between smooth principal bundles and transitive, locally trivial Lie groupoids. See locally trivial category for details.

One way to deal with this is to equip the 2-category with some structure of a homotopical category and allow morphisms of Lie groupoids to be anafunctors, i.e. spans of internal functors X←≃X^→YX \stackrel{\simeq}{\leftarrow} \hat X \to Y.

Regarded inside this wider context, Lie groupoids are identified with differentiable stacks. The (2,1)-category of Lie groupoids is then the full sub-(2,1)(2,1)-category of Sh(2,1)(Diff)Sh_{(2,1)}(Diff) on differentiable stacks.

An anafunctorX←≃C(U)→BGX \stackrel{\simeq}{\leftarrow} C(U) \to \mathbf{B}G from a smooth manifold XX to BG\mathbf{B}G is a Cech cocycle in degree 1 with values in GG, classifying GG-principal bundlePP.